Determination of Stiffness Coefficients at the Internal Vertices of the Tree Based on a Finite Set of Eigenvalues of an Asymmetric Second-Order Linear Differential Operator
Kanguzhin B. Kaiyrbek Z. Mustafina M.
August 2025Multidisciplinary Digital Publishing Institute (MDPI)
Symmetry
2025#17Issue 8
A second-order linear differential operator A is defined on a tree of arbitrary topology. Any internal vertex P of the tree divides the tree into (Formula presented.) branches. The restrictions (Formula presented.) of the operator A on each of these branches, where the vertex P is considered the root of the branch and the Dirichlet boundary condition is specified at the root. These restrictions must be, in a sense, asymmetric (not similar) to each other. Thus, the distinguished class of differential operators A turns out to have only simple eigenvalues. Moreover, the matching conditions at the internal vertices of the graph contain a set of parameters. These parameters are interpreted as stiffness coefficients. This paper proves that a finite set of eigenvalues allows one to uniquely restore the set of stiffness coefficients. The novelty of the work is the fact that it is sufficient to know a finite set of eigenvalues of intermediate Weinstein problems for uniquely restoring the required stiffness coefficients. We not only describe the results of selected studies but also compare them with each other and establish interconnections.
differential operators , Dirichlet conditions , Kirchhoff conditions , matching conditions , tree graphs
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Institute of Mathematics and Mathematical Modeling, Almaty, 050010, Kazakhstan
Institute of Mathematics and Mathematical Modeling
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